Abstract
Bayesian optimization (BO) based on the Gaussian process (GP) surrogate model has attracted extensive attention in the field of optimization and design of experiments (DoE). It usually faces two problems: the unstable GP prediction due to the ill-conditioned Gram matrix of the kernel and the difficulty of determining the trade-off parameter between exploitation and exploration. To solve these problems, we investigate the K-optimality, aiming at minimizing the condition number. Firstly, the Sequentially Bayesian K-optimal design (SBKO) is proposed to ensure the stability of the GP prediction, where the K-optimality is given as the acquisition function. We show that the SBKO reduces the integrated posterior variance and maximizes the hyper-parameters’ information gain simultaneously. Secondly, a K-optimal enhanced Bayesian Optimization (KO-BO) approach is given for the optimization problems, where the K-optimality is used to define the trade-off balance parameters which can be output automatically. Specifically, we focus our study on the K-optimal enhanced Expected Improvement algorithm (KO-EI). Numerical examples show that the SBKO generally outperforms the Monte Carlo, Latin hypercube sampling, and sequential DoE approaches by maximizing the posterior variance with the highest precision of prediction. Furthermore, the study of the optimization problem shows that the KO-EI method beats the classical EI method due to its higher convergence rate and smaller variance.
Highlights
Computer simulations are widely used to reproduce the behaviour of systems [1,2] through which their performance can be estimated
The first subsection demonstrates the Sequentially Bayesian K-optimal design for approximation problems, while the second one focuses on the comparison of the K-optimal enhanced Bayesian optimization problems
We proposed a simple acquisition function which is used to sequentially generate a design of experiments (DoE) which ensures the validity of Bayesian inference
Summary
Computer simulations are widely used to reproduce the behaviour of systems [1,2] through which their performance can be estimated. Surrogate models are introduced to represent the physical realities which can be computationally expensive and are difficult to obtain analytical solutions for. F is denoted as a response function of the real system with input x ∈ X ⊆ RD and observation y ∈ R which follows the form below: y = f ( x) + e. N i ∈ X ) and corresponding observations (Y ∈ R ), the surrogate models can be built to approximate f ( x) along with its statistics. The problem of proposing proper X is known as the Design of Experiments (DoE) and it was developed with various mathematical theories. DoE methods can be categorized as model-free and model-oriented
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